Planar Turán number of quasi-double stars

Given a graph H, we call a graph H-free if it does not contain H as a subgraph. The planar Turán number of a graph H, denoted by exP(n, H), is the maximum number of edges in a planar H-free graph on n vertices. A (h,k)-quasi-double star Wh,k, obtained from a path P₃=v₁v₂v₃ by adding h leaves and k leaves to the vertices v₁ and v₃, respectively, is a subclass of caterpillars. In this paper, we study exP(n,Wh,k) for all 1≤ h≤ 2≤ k≤ 5, and obtain some tight bounds exP(n,Wh,k)≤3(h+k)/h+k+2n for 3≤ h+k≤ 5 with equality holds if (h+k+2) n, and exP(n,W1,5)≤ 5/2n with equality holds if 12 n. Also we show that 9/4n≤ exP(n,W2,4)≤ 5/2n and 5/2n≤ exP(n,W2,5)≤ 17/6n, respectively.